Limit & continuity, B.Sc . 1 calculus , Unit - 1

This ppt covers following topic of unit - 1 of B.Sc. 1 Calculus :- Definition of limit , left & right hand limit and its example , continuity & its related example.

1. Shri Shankaracharya Mahavidyalaya JUNWANI,BHILAI NAGAR (C.G) TOPIC:-LIMIT AND CONTINUITY Presented by: Mrs. Preeti Shrivastava

2. CONTENT DEFINATION OF LIMIT. DEFINACTION OF LEFT HAND LIMIT. DEFINACTION OF RIGHT HAND LIMIT. EXAMPLE. CONTINUITY. (6) SOME EXAMPLES.

3. LIMITS DEFINACTION:- Let f ( x ) be a function of the variable x, then the value of f(x) at x=a, i.e. for some value `a’ of x is denoted by f(a). The value of f(a) may be of two kinds.

4. ( i ) A definite and finite quantity:- In this case the value of f(x) is said to be defined at x=a. (ii) An indefinite and indeterminate quantity:- In this case the value of f(x) is said to be undefined at x=a.

5. LEFT HAND LIMIT DEFINITION:- A function f(x) is sadi to tend to `l’ as a tends to a through the values less than a (or from left), if for any given there exists a such that. i.e. for every f(x)

6. Symbolicelly , we write:- Or Or f(a-0)= And is called the left hand limit (L.H.L.)

7. RIGHT HEND LIMIT DEFINITION:- A function f(x) is said to be tend to limit l as x tends to a, through the values greater than a (or form right), if for every there exist a such that i.e. for every Symbolically, we write:-

8. Or f(a+0) = And is called the right hand limit (R.H.L.) Ex:- If F(x)= when x<1 when x>1 Find if it exist.

9. SOLUCTION:- consider the L.H.L.

10. Again, consider R.H.L.

11. CONTINUITY:- The intuitive concept of continuity of a function is derived from its geometrical construction. If the graph of the function y=f(x) is curve which does not break at the point x=a then the function y=f(x) is called continuous at x=a. If the graph of the function break at some point then this point is called the point of discontinuity.

12. Ex:- IF F(x)= x -1 , x=-1 Is f(x) continuous at x=-1?

13. Soluction L:- R.H.L. at x=-1 F(-1+0)

14. L.H.L. at x= -1 F(-1-0)

15. Again when x=-1, then f(x)= -2 f(x)=2 Since f(-1-0)= f(-1) Hence the given function is continuous at x= -1

16. The following function for continuity at the origin. f(x)= , if x 0 , if x = 0 SOLUCTION:- Here f(0)=0 R.H.L. f(0+0)

17. L.H.L. = F(0-0) since f(0+0) = f(0),so f(x) is continuous at x=0.

18. THANK YOU